Differentials Definition and 236 Threads

Hello, I have been thought differential-calculus ages ago, but now when started reading some physics books (where infinitesimal quantities are used again and again) I realized I know nothing about calculus. I am unable to specify where exactly my problem lies, but I guess it lies in how to...

This is one of those questions I'd be afraid to ask, but here I go: If I have a quantity \Delta y= \Delta x+ (\Delta x)^2 + (\Delta x)^3 + (\Delta x)^4+\ldots and I let \Delta x tend to 0, and denote it with dx, is it correct to state that dy=dx ? If that is correct, then what's the reasoning...

First order nonlinear ODE -- Integrating factor + exact differentials, or not? Hello everyone, (I apologize if this did not format properly, if not I will attempt to edit it if that functionality is available upon submitting a question). I recently came across the following nonlinear ODE...

Wikipedia shows a proof of product rule using differentials by Leibniz. I am trying to correlate it to the definition of a differential and am having no success. Differential Definition: http://eom.springer.de/D/d031810.htm

Specifically, how do you prove the quotient rule using a similar method that Leibniz used for the product rule?: http://en.wikipedia.org/wiki/Product_rule#Discovery_by_Leibniz I've tried it once for d(u/v) but I keep getting a vdv term in the denominator.

hey there.. i really don't know how to start answering this question.. can someone please guide me to solve this question..

Claim 1: If \psi(x,0) has a compact support, and i\partial_t\psi(x,t) = -\partial_x^2 \psi(x,t), then \psi(x,t) does not have a compact support for any t>0. Claim 2: If \psi_1 and \psi_2 are the same in some environment of a point x_0, then \partial_x^2 \psi_1(x_0) = \partial_x^2...

Hi, I'm learning to do double integration by changing variables and wondering about this. Suppose we have f(x, y) and want to find the volume under the surface over some bounded area in the xy plane. Say, I want to change the variables into u and v by: u = 3x - 2y v = x + y I need to...

Homework Statement so, problem 1: xy' + 2y = 6x2y1/2 Homework Equations so this is a bernoulli, where, in the form y' + p(x)y = q(x)yn The Attempt at a Solution xy' + 2y = 6x2y1/2 y' + (2/x)y =6xy1/2 so in the form, p(x) = 2/x, q(x) = 6x, and v = y1/2 dividing both sides by...

My calculus book says sometimes derivatives can be regarded as the ratio of differentials, and sometimes they can't. Apparently, there's a similar rule for integrals. When can I think of derivatives and integrals as operations with differentials? And when can't I?

Homework Statement I have often heard that given say, ds/dt=k, that it is not entirely rigorous to say that ds=kdt. Why is that? If I view the derivative as nothing more than a ratio of the differentials, that seems perfectly reasonable. Also I see this done all the time, with acknowledgment...

Homework Statement The population of a country is divided in two groups: People who live in rural areas (R(t)) and people who live in urban areas (U(t)). People move from rural to urban areas with a rate m and from urban to rural areas with rate n. a) Introduce the fraction of...

Can someone please tell me necessary and sufficient conditions on a differential d \mathbf F, where \mathbf F is a vector field, for the differential to be exact?

To understand differentials better, I'm trying to use differentials dy and dx in the equation of the tangent line to the curve x^2 at point 3. Here is the equation of the tangent line to the curve x^2 at point 3: y=f'(3)(x-3)+f(3)=2(3)(x-3)+9=6(x-3)+9 But since we are dealing with the...

Homework Statement Find dy and evaluate dy for the given value x and dx . y = x^3 - 3x, x = 2, dx = 0.05 Homework Equations The Attempt at a Solution I have no idea how to begin this problem. I assumed that I find y'. Then plug in x into y'. Following that, I am not sure.

Homework Statement Use differentials to approximate the value of \sqrt{15} Homework Equations N/A The Attempt at a Solution I can't really figure out how to start this, so any help would be appreciated. I know how I'd do it by Newton-Raphson Method, but this is what the sample...

This is a simple dummy question. What are the conditions under which the following relationship holds, dx/dt = Inverse(dt/dx) = 1/(dt/dx) meaning if I want to do a derivative and I know t(x) but not x(t) when can I just calculate dt/dx and put it over 1 to get dx/dt. I see this in...

Homework Statement The period T of a simple pendulum with small oscillations is calculated from the formula T = 2pie (L / g)^1/2 . Suppose that measurements of L and g have errors of at most 0.5% and 0.1% respectively. Use differentials to approximate the maximum percentage error in the...

Homework Statement Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05cm thick to a hemispherical dome with diameter 50m Homework Equations A= 2(pi)r2 The Attempt at a Solution dr=0.0005 m r=25m dA=? (A=2(pi)r2)' dA= 4(pi)r*dr I won't go any further...

Homework Statement I have two problems from my differential equations assignment that I'm having difficulty with. I would appreciate some guidance! Homework Equations http://img10.imageshack.us/img10/3397/questionsk.th.jpg The Attempt at a Solution for no.10 I used reduction of...

Homework Statement Use differentials to estimate the amount of tin in a closed tin can with diameter 8cm, and height 12cm, if the tin is 0.04 cm thick. The way I did this was to subtract the Max volume by the Enclosed volume. Max Volume is pi(4)(12) Enclosed Volume is pi ( 8 - 2(0.04) )/2...

Homework Statement Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a hemispherical dome with diameter 54 m. Homework Equations dy=dy/dx *dx Surface area of a hemishpere=2pi*r^2 The Attempt at a Solution dy=4pi*r dx dy=4pi*27...

I posted a question at another forum about "choked flow" where a fluid (in this case air) if flowing through a very small orifice. Apparently when the pressure difference on each side of a orifice is about 2:1 and above then the speed of the air passing through the orifice is limited to the...

I am very interested in math and I find calculus to be a particularly interesting subject, but one major problem I have with it is that I cannot find a consistent explanation of the rules of differentials (infantesimals) that explains all the things mathematicians do with them. I have truly...

ok I want to clear my broad views upon calculus.. first of all I wuz taught calculus 2-3 years back with differentials and Integration.. and ever since iam not really interested in doing this because I want to know wot iam doing ? I want to know how do they occur naturally ? I mean like i...

Homework Statement How can I find out if a function is a solution of a wave equation such as: (a) xt (b) log(xt) (c) x² + c²t² The Attempt at a Solution Is it simply differentiating the funtion with respect to 'x' twice and equating this to the product of 1/c² and...

The air in a room with volume 180 m^3 contains 0.20% carbon dioxide initially. Fresher air with only 0.05% carbon dioxide flows into the room at a rate of 3 m^3/min and the mixed air flows out at the same rate. (a) Find the percentage of carbon dioxide in the room as a function of time...

Homework Statement Verify that the given function is in the kernel of L. y(x)=x-2 L = x2D2 + 2xD - 2 Homework Equations The Attempt at a Solution I took the first and 2nd derivative of y(x), and got y'(x)= -2x-3 y''(x)= 6x-4 Then plugged it into L (and a little simplifying) and got...

Hello. Can someone please explain why I have to transform an integral of a differential function into the form Integral ( lnx 1/x dx ) for example, for Integral ( lnx ). It seems to only be done with transcendental functions and not the algebraic ones... ie. Integral ( x^2 ) != Integral...

Homework Statement Solve: (2t+x) dx/dt + t = 0Homework Equations y' +p(X)y = q(x) and y(x) = (\intu(x)q(x) + c)/u(x) where u(x) = e\intp(x)dx Note this u(x) is 2 to the power of the integral of p(x) The Attempt at a Solution (2t+x) dx/dt + t = 0 becomes: dx/dt + t/(2t+x) = 0 by dividing...

Homework Statement I'm looking for \frac{\partial{P}}{\partial{V}} at fixed T and fixed S. Homework Equations P=\frac{TS}{4V} The Attempt at a Solution...

Homework Statement Use differentials to estimate the amount of paint needed to apply a coat of paint 0.03 cm thick to a hemispherical dome with diameter 54 m. (Round the answer to two decimal places.) Homework Equations V = 1/2(4/3(pi*r^3)) = 1/2(4/3(pi*(1/2D)^3)) = 1/2(4/3*pi(1/8D^3)) =...

I am trying to understand a derivation posted by Pervect a long time ago (I suspect Pevect is no longer active) that involves differentiation and I was hoping someone here could fill in some of the steps to make it clearer. The original posts by Pervect are here...

1. The problem Use Differentials to approximate (24.5)^(1/2) + (9.5)^(1/2). Compare your answer to your calculator's answer. Homework Equations I used z = (x)^(1/2) + (y)^(1/2) The Attempt at a Solution What I used: let z = (x)^(1/2) + (y)^(1/2) x = 25 y = 10 dx = 0.5 dy = 0.5...

Homework Statement Consider the Atwood's machine of Lecture 8. We wish to use this machine to measure our local acceleration of gravity with an accuracy of 5% [i.e. (Delta g)/g = 0.05]. To begin, suppose we let the mass m_1 fall through a distance L. 3.1 Find an expression for the...

Hi everyone! I am a UK grad student working in acoustics. My own background is in EE, so i am largely self taught in thermodynamics. Consequently id really appreciate any insight any of you real physicists can give me with my problem! Following derivations in books i can derive the enthalpy...

total differentials! [SIZE="4"]a.) f = 1/(x+y2) [SIZE="4"]b.) f = ln(x2y) [SIZE="4"]c.) P = [(RT)/(V-B)]-[A/(T1/2V2)] what are the total differentials of these functions?

Can someone explain to me why \int\frac{x}{x^2+1} = \frac{1}{2}ln(x^2+1) WHERE DOES THE ONE HALF COME FROM ?

Homework Statement Use differentials to estimate the amount of tin in a closed tin can with diameter 8 cm and height 12 cm if the tin is .04 cm thick. Homework Equations dz = (dz/dx) dx + (dz/dy) dy The Attempt at a Solution To find the area of the tin can we can see it as a...

Homework Statement This isn't exactly HW problem , but a "simple" worked problem that is supposed to illustrate exact differentials... Consider a differential dZ =2xy⋅dx+x^2dy integrated on two paths where Path I is x=y and Path II is x^2=y. The integrations are from (x,y)=(0,0) to...

Homework Statement y" + 2y' + 10y=0 y(0)=1 y'(0)= -1 solve initial value problem Homework Equations e^( ~ + iu)t= e^ ~t (cos ut + i sin ut) The Attempt at a Solution i've gotten pretty far into the problem, but i just can't seem to get the correct final answer. I changed y" +...

If you have a water tank with an inflow u, and an outflow v, you have that \frac{dV}{dt} = A \frac{dh}{dt} = u - v. You can now linearize this expression so that you get A \frac{d}{dt}(\Delta h) = \Delta u - \Delta v = (u-u_0) - (v-v_0), where \Delta h = h-h_0. I think I...

often times you will see naive people cancel differentials to obtain whatever it is they want for example : \frac{dx}{dy} \frac{dy}{dz} = \frac{dx}{dz} now i know this isn't rigorous and my question is actually about partials. is there ever an occassion/space/set/etc where i can do...

Hello, I'm toying around with a Jacobian that has raised some interesting problems. It's a case of differentiating rates of some variable x, with respect to itself. First one I suspect the answer is zero, though perhaps my reasoning is a bit flawed. 1. \frac{d}{d\theta}(\dot{\theta})...

The proper time is defined by d\tau^2=g_{\mu\nu}dx^\mu dx^\nu Suppose we have flat space time with one space dimension. d\tau=\sqrt{dt^2-dx^2} =dt\sqrt{1-\frac{(dx^2)}{(dt^2)}} =dt\sqrt{1-\left(\frac{dx}{dt}\right)^2} Can this be rigorous?

[SOLVED] Derivatives and differentials Homework Statement Hmm, when I have \frac{dx^2}{dx}, does this equal zero or 2x? What confuses me is the way it is written.

Hello, I'm having trouble with this maths question and was wondering if someone could help me?? The question asks: The electrical resistance R of a wire is given by k / r^2 where k = constant, r = radius of wire. Use differentials to estimate the percentage error in the...

Homework Statement I hate infinitesimals and differentials. When I learned calculus, we used Liebniz notation df/dx only as a convenience for using the chain rule. In physics, apparently, people just play around with differentials and infinitesimals and expect to get the right answer...

[SOLVED] Gravitation Force Using Differentials Homework Statement Three part question: 1. Consider a solar system similar to our Sun and Earth, where the mass and radius of the planet are 4.22e24 kg and 6.63e6 m, respectively, the mass of the sun is 2.08e30 kg and the planet-sun...

[SOLVED] Proving a Fact About Exact Differentials Problem. Let D be a disc and let P and Q be functions on D with continuous partial derivatives with respect to x and y. Let C be any closed curve in D. Given the fact that \int_C (P \, dx + Q \, dy) = 0 \leftrightarrow Q_x = P_y show...